impatience and Accumulation

In studying the concept of impatience Koopmans [8] was led to a useful class of preference orderings which generalises the familiar class of additively separable preferences. These aggregator preference orderings (see Definition 2.3) exhibit a richer variety of behaviour for the various concepts of impatience that can be associated with preference orderings among commodity sequences over time and provide a formal framework for the analysis of impatience along the lines of Irving Fisher [6, Chap. IV]. Beals and Koopmans [ 1 ] studied the paths of capital accumulation that arise from such preference orderings in an economy with a single (capital) good, a simple recursive technology set and a single representative consumer. They showed that if the rate of impatience (Definition 2.5) decreases whenever consumption in the first period is increased (Assumption BS’), then the optimal path of capital accumulation has a simple asymptotic behaviour-it converges to a steady state that depends only on the rate of return (Eq. (3.1)) associated with the constant path from the given initial condition. Our objective is to provide an extension of this earlier analysis.’ Section 2 lays out the basic assumptions on technology and preferences. In Section 3, after establishing in a natural abstract setting the basic continuity properties of an optimal path (Theorem 3.1) we show that an alternative condition (Assumption 5) leads to the same asymptotic behaviour (Lemma 3.3-Theorem 3.8). This restriction on impatience expresses in an alternative form Fisher’s idea that an increase in consumption in early periods (the first two periods) reduces impatience.